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Electrostatic solution of massless quenches in Luttinger liquids
by Paola Ruggiero, Pasquale Calabrese, Thierry Giamarchi, Laura Foini
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Submission summary
Authors (as registered SciPost users):  Laura Foini · Thierry Giamarchi · Paola Ruggiero 
Submission information  

Preprint Link:  https://arxiv.org/abs/2203.06740v3 (pdf) 
Date accepted:  20220921 
Date submitted:  20220522 18:54 
Submitted by:  Ruggiero, Paola 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The study of nonequilibrium dynamics of manybody systems after a quantum quench received a considerable boost and a deep theoretical understanding from the path integral formulation in imaginary time. However, the celebrated problem of a quench in the Luttinger parameter of a one dimensional quantum critical system (massless quench) has so far only been solved in the realtime Heisenberg picture. In order to bridge this theoretical gap and to understand on the same ground massive and massless quenches, we study the problem of a gaussian field characterized by a coupling parameter K within a strip and a different one K0 in the remaining two semiinfinite planes. We give a fully analytical solution using the electrostatic analogy with the problem of a dielectric material within a strip surrounded by an infinite medium of different dielectric constant, and exploiting the method of charge images. After analytic continuation, this solution allows us to obtain all the correlation functions after the quench within a path integral approach in imaginary time, thus recovering and generalizing the results in real time. Furthermore, this imaginarytime approach establishes a remarkable connection between the quench and the famous problem of the conductivity of a TomonagaLuttinger liquid coupled to two semiinfinite leads: the two are in fact related by a rotation of the spacetime coordinates.
Author comments upon resubmission
We are submitting a revised version of the manuscript
“Electrostatic solution of massless quenches in Luttinger liquids”.
We would like to thank the editors for their work and the referees for
their useful comments and suggestions.
We believe that we thoroughly addressed all the comments of the referees
in the new version of the manuscript.
Sincerely,
Paola Ruggiero
Pasquale Calabrese
Thierry Giamarchi
Laura Foini
List of changes
 Few typos corrected
 New section added (Section 7)
 Last sentence in Appendix added
Published as SciPost Phys. 13, 111 (2022)
Reports on this Submission
Anonymous Report 3 on 2022830 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2203.06740v3, delivered 20220830, doi: 10.21468/SciPost.Report.5605
Strengths
The paper fills a gap compared to previous works, by proposing to use the imaginarytime path integral for massless quenches in Luttinger Liquids
It makes clear the analogy with the spatial inhomogeneous Luttinger liquid, obtained by a permutation of space and time coordinates.
Weaknesses
The paper recovers an already know result (Eq.29) obtained in Refs. 29,5052. It is not so clear to guess whether the different method exposed here could give access to other unsolved problems.
Report
The paper is based on the path integral formulation for massless quenches in Luttinger liquids. The main interest of the paper resides into an alternative method to those used in Refs. 29,5052 to recover the same result. The starting formal expression that allows to switch from real to imaginary time , given by Eq. 4, corresponds to Eq.2 in Ref.5 for instance, where it was developed for massive quenches. Its limitations was addressed in Ref. 6, page 4: "Unfortunately, in confined geometries, only a few field theories with specific boundary conditions can be solved analytically in such a way to have results that can be continued from real to complex values". If I understand well, the present situation corresponds to such an example, thus could be valuable in this respect (maybe the authors could confirm or not this point?). Nonetheless, I have hesitated as it is not clear whether the present method could solve other problems such as those mentioned in the conclusion. The pedagogical value of the paper incites me to recommend its publication in SciPost. The connexion to spatially inhomogeneous Luttinger liquid is also interesting.
Requested changes
I have some remarks about the way similarity to the spatially inhomogeneous Luttinger liquid is presented. Either in the introduction or section 6, authors insist only on the motivation and result for the dc conductance, while the finite frequency Green's function, related to the nonlocal conductivity, has been obtained (first) in Ref. 38 (and not in Ref.39). The present equation 39 corresponds precisely to Eq.9 in Ref.38. It would be interesting to provide the interpretation thus given of this equation, due to multiple reflexions inside the wire which acts a a FabryPerot resonator. By developing the scattering approach to plasmons, the ac conductance was related exactly to the transmission coefficient constructed through evolution of plasmonic modes. As the sum of all transmitted spikes sums up to one (and the sum of reflected ones sum up to zero), the dc conductance is equal to e^2/h$. By the way, it is stated in the introduction (page 2) that "as soon as we consider an inhomogeneous Luttinger parameter K(y,z)... the propagator has to be determined numerically". This is not clearly the case in Refs.38,53,54 which have provided an analytical solution for the propagator.
Anonymous Report 2 on 202265 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2203.06740v3, delivered 20220605, doi: 10.21468/SciPost.Report.5184
Report
I thank the authors for their reply to my questions about the manuscript. As stated in my previous report, the manuscript is scientifically correct and wellwritten. It elaborates on the study of the massless quench in Luttinger liquids, which has a broad interest for the community working on the nonequilibuim dynamics after a quench. On a more technical side, it gives an exact analytic solution for the Green's function in an inhomogeneous medium. Despite the simple form of the inhomogeneity of the Luttinger parameter, this result is quite remarkable. For these reasons, I recommend this manuscript for the publication in SciPost Physics.
Report
I thank the authors for their detailed response. Unfortunately they appear to have misunderstood my main point, which was not that the whole approach using imaginary time is narrow and pedestrian, but that this particular contribution, although correct, is so, and is not at the level suitable for publication in SciPost.